Matrix algebra ib1

8/7/2019 Matrix algebra ib1

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MATRIX ALGEBRAUnit 4 IB SL Math


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MATRIX: A rectangular

arrangement of numbers in

rows and columns.

The ORDER of a matrix isthe number of the rows and

columns.

The ENTRIES are the

numbers in the matrix.

-

502

126

rows

columns

This order of this matrixThis order of this matrix

is a 2 x 3.is a 2 x 3.


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-

67237

89511

36402

-

3410

200

318 ? A0759

-

20

11

-

6

0

7

9

3 x 3

3 x 5

2 x 2 4 x 1

1 x 4

(or square

matrix)

(Also called a

row matrix)

(or square

matrix)

(Also called a

column matrix)


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To add two matrices, they must have the sameorder. To add, you simply add correspondingentries.

-

-

34

03

12

70

43

35

-

!

)3(740

0433

13)2(5

-

!

44

40

23


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-

9245

3108

-

2335

2571

? A)1(8 70 51 23

55 34 32 )2(9 =

= ?7 7 4 50 7 5 7 A


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To subtract two matrices, they must have theTo subtract two matrices, they must have thesame order. You simply subtract correspondingsame order. You simply subtract correspondingentries.entries.

-

-

232

451704

831

605429

-

!

2833)2(1

)4(65015

740249

-

!

603

1054

325


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-

-

724

113

810

051

708

342

? ?A=5-2

-4-1 3-8

8-3 0-(-1) -7-1

1-(-4)

2-0

0-7

=

2 -5 -5

5 1 -8

5 3 -7A


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In matrix algebra, a real number is often called aSCALAR. To multiply a matrix by a scalar, you multiplyeach entry in the matrix by that scalar.

-

14

024

-

!

416

08

-

!

)1(4)4(4

)0(4)2(4


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-

-

86

54

30

212

-

!

)8(360

52412

!

? ?? A

AA-2

6

-3 3

-2(-3)-5

!!

-2(6) -2(-5)

-2(3) 6 -6

-12 10


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Equal MatricesTwo matrices are equal if the entries in corresponding positions

are equal

-

{

-

-

!

-

10

47

10

74

05.

23

0

2

123


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1-11

MULTIPLICATIONOF MATRICES


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1-12


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1-13

EXAMPLE 5 CONTINUED


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1-14

EXAMPLE 5 CONTINUED


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NOTE

CW

For 2x2 (two by two) matrices A and B then AB { BA


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45

23

Notice the different symbol:

the straight lines tell you to

find the determinant!!

(3 * 4) - (-5 * 2)

12 - (-10)

22

=

45

23

Finding Determinants of Matrices

=

=


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241

212

21

-1

-2

4

= [(2)(-2)(2) + ( )( )(-1) + ( )(1)(4)]

[( )(-2)(-1) + (2)( )(4) + ( )(1)(2)]

[-8 + +12]

-

- [6 + 40 + 0]

4 6- 40

Finding Determinants of Matrices

=

= = -42


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-

10

01

Identity matrix: Square matrix with 1s on the diagonal and zeros

everywhere else

2 x 2 identity matrix

-

100

010

001


The identity matrix is to matrix multiplication as ___ is to regularmultiplication!!!!1

Using matrix equations


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-

10

01

Identity matrix: Square matrix with 1s on the diagonal and zeros

everywhere else


-

100

010

001


The identity matrix is to matrix multiplication as ___ is to regularmultiplication!!!!1



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Multiply:

-

10

01

-

43

25=

-

43

25

-

1001

-

4325

=

-

4325

So, the identity matrix multiplied by any matrix

lets the any matrix keep its identity!

Mathematically, IA = A and AI = A !!


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InverseMatrix:


2 x 2

-

dc

ba

In words:

Take the original matrix.Switch a and d.

Change the signs of b and c.

Multiply the new matrix by 1 over the determinant of the original

-

ac

bd

bcad

1!1A

!A


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-

24

410

)4)(4()10)(2(

1

-

24

410

4

1=

-

2

11

125


Example: Find the inverse of A.

-

104

42!A

!1A

!1

A


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Find the inverse matrix.

-

25

38

Det A =8(2) (-5)(-3) = 16 15= 1

Matrix A

Inverse =

-

det

1Matrix

Reloaded

-

85

321

1= =

-

8532


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What happens when you multiply a matrix by its inverse?

1st: What happens when you multiply a number by its inverse?717 y

A & B are inverses. Multiply them.

-

85

32=

-

25

38

-

10

01

So, AA-1 = I


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Why do we need to know all this? To Solve Problems!

Solve for Matrix X.

=

-

2538 X

-

1314

We need to undo the coefficient matrix. Multiply it by its INVERSE!

-

8532 =

-

2538 X

-

8532

-

1314

-

10

01

X=

-

34

11

X =

-

34

11


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You can take a system of equations and write it with

matrices!!!

3x + 2y = 11

2x + y = 8becomes

-

12

23

-

y

x=

-

8

11

Coefficient

matrix

Variable

matrix

Answer matrix



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Let A be the coefficient matrix.

Multiply both sides of the equation by the inverse of A.

-

!

-

-

!

-

-

!

-

8

11

8

11

8

11

1

11

Ay

x

Ay

xAA

y

xA

-

12

23 -1=

-

32

21

1

1=

-

32

21

-

32

21

-

12

23

-

y

x=

-

32

21

-

8

11

-

1001

-yx =

-

25

-

y

x=

-

2

5


-

12

23

-

y

x=

-

8

11Example: Solve for x and y .

!1A


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Wow!!!!

3x + 2y = 11

2x + y = 8

x = 5; y = -2

3(5) + 2(-2) = 11

2(5) + (-2) = 8

It works!!!!


Check:


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You Try

Solve:

4x + 6y = 14

2x 5y = -9

(1/2,2)

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Matrix algebra ib1